$\eta$-symbols in exceptional field theory

We present the universal form of $\eta$-symbols that can be applied to an arbitrary $E_{d(d)}$ exceptional field theory (EFT) up to $d=7$. We then express the $Y$-tensor, which governs the gauge algebra of EFT, as a quadratic form of the $\eta$-symbols. The usual definition of the $Y$-tensor strongly depends on the dimension of the compactification torus while it is not the case for our $Y$-tensor. Furthermore, using the $\eta$-symbols, we propose a universal form of the linear section equation. In particular, in the SL(5) EFT, we explicitly show the equivalence to the known linear section equation.

In order to make contact with the conventional supergravity in d-dimensions, it is useful to decompose the generalized coordinates into the physical coordinates x i (i = 1, . . . , d) and the dual coordinatesx i ; (x I ) = (x i ,x i ) . By introducing the O(d, d) T -duality-invariant metric, the consistency condition of DFT, the so-called the section condition, is expressed as Here, ⊗ represents that are satisfied for arbitrary fields or gauge parameters A and B. Under the section condition, the gauge algebra generated by the following generalized Lie derivative is closed: As a natural generalization of DFT, the E d(d) exceptional field theories (EFTs) [8][9][10][11][12][13][14][15][16] have been formulated in a manifestly E d(d) U -duality covariant manner (see [17][18][19][20] for the initial attempts). In EFT, the generalized coordinates x I (I = 1, . . . , D) are defined to transform in a fundamental representation, called the R 1 -representation (see Appendix A.2). The generalized Lie derivative is defined bŷ where the Y -tensor Y IJ KL for each d (4 ≤ d ≤ 7) is given as follows [11] (see [21][22][23] for the generalized Lie derivative in the context of exceptional generalized geometry): SL ( Here, e.g., γ IJ A is the gamma matrix for the SO (5,5) group and d IJK is the totally symmetric tensor intrinsic to the E 6(6) group (see Appendix B for the details of these d-dependent tensors).

A sketch of the basic idea
The section condition in DFT has been proposed on the basis of the level-matching constraint in string sigma model [1][2][3], In the canonical formulation, the level-matching constraint, or the momentum constraint H σ = 0, can be expressed as where P i (σ) are the conjugate momenta to X i (σ) and Z I (σ) are the generalized momenta, By supposing that the operator L V ≡ dσ V I X J (σ) Z I (σ) , (2.4) acts as the generator of the diffeomorphism along V I ∂ I , we can roughly identify Z I with ∂ I , and the momentum constraint corresponds to the section condition in DFT, η IJ ∂ I ⊗ ∂ J = 0 .
A similar consideration has been given for M-theory branes in [26,27]. In the case of an M2-brane wrapped on a 4-torus, the momentum constraint H A = 0 (A = 1, 2: index for spatial coordinates on the M2-brane) is rewritten as where Again by supposing the generalized momenta Z I to act as ∂ I ≡ ∂/∂x I with (x I ) = (x i , , the momentum constraint is expressed as the section condition, Similarly, in the case of an M5-brane wrapped on a 5-torus, the momentum constraint, H A = 0 (A = 1, . . . , 5), has been expressed in a bilinear form (see [27] for the details), a k η IJ; k Z I Z J + b k 1 ···k 4 η IJ; k 1 ···k 4 Z I Z J = 0 , (2.8) where the matrices η k ≡ (η IJ; k ) and η k 1 ···k 4 ≡ (η IJ; k 1 ···k 4 ) have the form: Here, a k and b k 1 ···k 4 (= b [k 1 ···k 4 ] ) are arbitrary constants and the section conditions can be decomposed into two parts, η IJ; k ∂ I ⊗ ∂ J = 0 , η IJ; k 1 ···k 4 ∂ I ⊗ ∂ J = 0 . (2.10) The former condition is the same as the section condition coming from the M2-brane and the latter is intrinsic to the M5-brane.
A similar consideration for a Dp-brane in type II string theory was made in [28] (though the U -duality covariance is not manifest there), and the general rule we observe is that each p-brane provides the corresponding η-symbol η k 1 ···k p−1 and the associated section condition η IJ; k 1 ···k p−1 ∂ I ⊗∂ J = 0 . In fact, a set of multiple indices with one dimension fewer in the spatial dimension of branes is known to form the string multiplet of E d(d) group. The dimension of the string multiplet for each U -duality group is given as follows (see [29] for a concise review): duality group SL(5) SO(5, 5) E 6(6) E 7 (7) dimension of string mult. 5 10 27 133 . (2.11) As is clear from the dimension, the string multiplet is the same as the R 2 -representation that determines the section condition [29] (see also Appendix A.2). Now, it is natural to expect that each brane in the string multiplet provides a particular η-symbol and the corresponding section condition, and the sum of all these section conditions is equivalent to the section condition in the E d(d) EFT. We thus introduce the following set of η-symbols associated with branes in the string multiplet in M-theory and type IIB theory: , · · · , , η m 1 ···m 6 , n 1 n 2 α √ 6! 2! , η m 1 ···m 6 , n 1 ···n 6 α √ 6! 6! 1 6 4 /1 6 3 , · · · , (2.12) where the multiple indices are totally antisymmetrized and the ranges of the indices are k, l = 1, . . . , d, m, n = 1, . . . , d − 1, and α, β = 1, 2. Each η-symbol corresponds to a brane specified below the underbrace (see [29] and also [30] for the notation of exotic branes b c n ). The ellipses are relevant only for the E d(d) EFT with d ≥ 8, which is not considered here.
The above set of η-symbols would be essentially the same as the set of the η-symbols introduced in an "F-theory" [31][32][33][34][35]. 1 There, the η-symbols were introduced as the Clebsch-Gordan-Wigner coefficients connecting R 1 ⊗ R 1 and the R 2 -representation, and the Virasorolike constraint was expressed as η IJ; I P I P J = 0 . (2.13) The generalized Lie derivative was obtained from the Virasoro-like constraint, and by comparing with the generalized Lie derivative, the Y -tensor in EFT was expressed as , (2.14) 1 The set of η-symbols was introduced in [22] as the projection ×N : E × E → N , and in [36] as the wedge product ∧ : R1 ⊗ R1 → R2. Note that the wedge product is defined for more general representations.
where the singlet constraint Ω IJ ∂ I ⊗ ∂ J = 0 was introduced for d = 7 from consistency with the EFT. The explicit form of the η-symbol was found in [34] using a different convention from ours.
In this paper, instead of attempting to translate the η-symbols found in [34] into our convention, we utilize the linear map considered in [30]. As has been well known [12,37], EFT can reproduce both M-theory and type IIB theory. Depending on which theory one has in mind, there are two natural parameterizations of the generalized coordinates: x I for M-theory and x M for type IIB theory. The linear map in [30] provides a relation between the two parameterizations: When we consider the linear map, we decompose the physical coordinates x i (i = 1, . . . , d) for M-theory and x m (m = 1, . . . , d − 1) for type IIB theory as Here, x z in the M-theory side corresponds to the coordinate on the M-theory circle. If we adopt the type IIA picture (by compactifying the M-theory circle), the linear map corresponds to a single T -duality along the x y or x y directions in type IIA/IIB theory. Indeed, in [30], it was shown that the linear map between two generalized metrics, M IJ (M-theory) and M MN (type IIB theory), precisely reproduces the well-known T -duality transformation rules for supergravity fields. In this paper, we apply this linear map to η-symbols in M-theory/type IIB theory.
To be more specific, following the convention used in [30], we parameterize the generalized coordinates as M-theory: where the coordinates other than the physical coordinates are winding coordinates associated with some branes specified below the underbrace and ellipses again are relevant only for the In the above parameterized generalized coordinates x I for M-theory, we begin by considering two η-symbols, which are trivial extensions of the η-symbols associated with M2-/M5-branes shown in (2.9).
Under a compactification on the M-theory circle, an M2-brane becomes a D2-brane or an F-string in type IIA theory, and under a T -duality, it can become a D1/D3-brane or an Fstring. Correspondingly, under the linear map, the η-symbol (η k ) = (η a , η α ) can be mapped to an η-symbol, η ay or η α , associated with a D3-brane or an F/D-string in type IIB theory.
Similarly, the η-symbol (η k 1 ···k 4 ) = (η a 1 ···a 4 , η a 1 a 2 a 3 α , η a 1 a 2 yz ) can be mapped to an η-symbol, η a 1 ···a 4 y, y , η a 1 ···a 3 y α , or η a 1 a 2 , associated with a Kaluza-Klein monopole (KKM), an NS/D5brane, or a D3-brane in type IIB theory. Repeating the linear map, we can find almost all of the η-symbols described in (2.18). The only η-symbols that cannot straightforwardly be obtained from the linear map are η [k 1 ···k 6 , l] and η [m 1 ···m 5 , n] , which correspond to 8-branes in M-theory and 7 2 -branes in type IIB theory, respectively. These branes are not related to other branes described in (2.18) via T -duality transformations, but they are related to each other.
In fact, by requiring the SL(d) or SL(d − 1) covariance in the M-theory or type IIB theory sides, they also can be determined completely. Then, we find the explicit form of all η-symbols (and also the Ω-tensor) and can construct the Y -tensor through (2.14).
We expect that, in the same manner as [26][27][28], all of the η-symbols obtained in this paper will also be read off from the momentum constraint (i.e., Virasoro-like constraint) in worldvolume theories of branes appearing in (2.12), but we leave the task for future work, and here we will concentrate on the determination of the η-symbols for the E d(d) EFT (d ≤ 7).

Explicit form of η-symbols
In this section, we begin by showing the explicit matrix form of η-symbols in two generalized coordinates, x I and x M , associated with M-theory and type IIB theory, respectively. Their derivations are explained in section 3.3. In Appendix B, we explain how to reproduce the known Y -tensors from our η-symbols.

M-theory parameterization
When we adopt the M-theory description, the decomposition of η I becomes The explicit forms of each matrix, η I = (η IJ; I ), are Here, η k 1 ···k 6 , l KKM is defined to satisfy η [k 1 ···k 6 , l] KKM = 0. We also define the η-symbols η I = (η IJ; I ) as For example, η k is defined as (3.10) The position of the indices is converted but η I and η I have the same components as a matrix.
In the above expressions, we are supposing the case of the E 7(7) EFT but the expressions for the E d(d) EFT with d ≤ 6 can be obtained via a simple truncation of the above matrices.
For example, in the SL(5) EFT (where d = 4), the generalized coordinates are given by ) and the non-vanishing η-symbols become The number of the generalized coordinates, the η-symbols, and the corresponding brane charges for d ≤ 7 can be summarized as follows: , y i 1 i 2

Type IIB parameterization
When we adopt the type IIB description, we consider the following decomposition of η-symbols: where the matrices take the form We also define the η-symbols η M = (η MN; M ) as A list of non-vanishing coordinates and η-symbols for each d is , y α m F1/D1 [6] , y m 1 m 2 m 3 , y α m F1/D1 [8] , y m 1 m 2 m 3

The linear map
We utilize the following linear map between generalized coordinates x I (M-theory) and x M (type IIB) [30]: Under the linear map, e.g., the matrix η a = (η IJ; a ) associated with an M2-brane is mapped to a matrix η ay = (η MN; ay ) associated with the D3-brane in type IIB theory, where the minus sign is introduced by convention. Similarly, we can relate all of the η-symbols for M-theory and type IIB theory via the linear map S. By introducing a transformation matrix for the R 2 -representation T M J , we can express the linear map for the η-symbols as Here, the matrix T that maintains the SL(d − 2) covariance can be summarized as follows: In the case of d = 7, as we mentioned in section 2, we cannot determine the matrix form of η [k 1 ···k 6 , l] = η k 1 ···k 6 l and η [p 1 ···p 5 , q] = η p 1 ···p 5 q only through the above mapping procedure.
Assuming that these matrices are symmetric and constructed only from the Kronecker deltas, the possible form for d ≤ 7 is From these ansatz, we define Supposing that η k 1 ···k 6 , l and η p 1 ···p 5 , q are related with each other by the linear map, under the decomposition {i} → {a, α} and {m} → {a, y}, the non-trivial components η a 1 ···a 5 , y and η a 1 ···a 4 y, c (which include the contribution from η k 1 ···k 6 l ) should be expanded with η k 1 ···k 6 , l in the following general forms (3.52) This requires χ 1 = 3 χ 2 and χ 3 = 2 χ 4 . We can determine the overall constant (up to sign) of η k 1 ···k 7 and η p 1 ···p 6 (i.e. χ 2 and χ 4 ) by further requiring the conditions By choosing a sign convention, we obtain the η-symbols shown in the previous subsections.
Similarly, we can also determine the matrix form of the Ω-tensors that appear in d = 7.
Supposing that they are also constructed from combinations of products of Kronecker deltas, the defining properties, require them to have the following form up to the overall sign convention: In order for the Ω-tensor in the type IIB side, namely Ω MN ≡ (S −1 ) M I Ω IJ (S −T ) J N , to be expressed covariantly by means of the Kronecker deltas, we shall choose the upper sign, and then the Ω-tensor in the type IIB side becomes (3.39). In this manner, we have determined all of the η-symbols and the Ω-tensor.

Properties of η-symbols
We can easily check that the identities In the case of E 7(7) EFT, we can check additional identities. If we define we can show a relation that connects the two types of η-symbols, η IJ; I and η IJ; I , which has the eigenvalues 70 "+1" and 63 "−1." The same relations are also satisfied in the type IIB side, and there the matrix K = (K MN ) becomes In fact, t I corresponds to the generators of the E 7(7) group. By using the generators of the E 7(7) group shown in Appendix A.2, t I can be expressed as Similarly, the t M are related to the generators in the type IIB parameterization as Then, t [k 1 ···k 6 , k] or t [p 1 ···p 5 , p] and t p 1 ···p 6 (12) are Cartan generators, and the matrix K corresponds to the Cartan-Killing form.
We can also check the following identities [14]:

Generalized Lie derivative
By using the obtained η-symbols, the section condition η IJ; I ∂ I ⊗ ∂ J = 0 can be expressed as follows (see [22] for a quite similar section condition and also [25] for a section condition in the "underlying EFT"): where c 1 and c 2 are defined in (3.48). In particular, when we consider, e.g., the SL(5) EFT, the above section conditions are truncated easily to get The section condition in the type IIB parameterization, η MN; M ∂ M ⊗ ∂ N = 0, can also be rewritten in a similar manner, though we will not show this explicitly.
There are two well-known solutions to the section condition. One is the solution, called the M-theory section, where are satisfied. In the type IIB section, fields depend only on the d − 1 coordinates x m (see [37] for the type IIB section in the SL(5) EFT and also [12][13][14]16] for a higher E d(d) EFT).
On the M-theory section, the generalized Lie derivative reduces to the exceptional Dorfman bracket [21][22][23]38]. Indeed, by using the Y -tensor, and the explicit form of the η-symbols and the Ω-tensor, we obtain In the last line, we have repeatedly used the Schouten-like identities such as In fact, a condition, is necessary for the closure of the gauge algebra [11], and for d ≤ 7, it is indeed satisfied under the section condition (1.7) [11]. Therefore, a gauge parameter of the form V I = η IJ; I ∂ J f I is a generalized Killing vector for an arbitrary f I . Moreover, V I = Ω IJ χ J with χ J satisfying is also a trivial generalized Killing vector [14], where the identity (3.65) is used in the second equality and (4.15) is used in the last equality.
On the other hand, if we choose the type IIB section, the generalized Lie derivative takes the form  We can easily show that this leads to the section condition, In fact, in the context of generalized geometry, essentially the same set of generalized vectors has been considered in [39] (see also [40]). There, the maximal null subspace has been called the Dirac manifold or the Dirac structure, and the set of generalized vectors λ a has been called the basis representation of the Dirac structure. In addition, it has been shown that the Dirac structure can be characterized by an antisymmetric tensor, which is denoted by β ij here. Alternatively, we can also characterize the Dirac structure by using a pure spinor [41].
A linear differential equation similar to (5.1), which reproduces the section condition, is called the linear section equation in [11]. There, a linear section equation in E d(d) EFT for d ≤ 7 was proposed, but the equation strongly depends on the dimension d and it becomes complicated for higher d. Here, using the η-symbols, we propose a simple linear section equation, and show that it is equivalent to the proposal of [11] for the SL(5) EFT.
Our linear section equations take the form where λ a (a = 1, . . . , N ) is a set of generalized vectors satisfying the null conditions If we show all of the indices explicitly, the linear section equations in the M-theory/type IIB parameterization become M-theory : λ a I η IJ; I ∂ J = 0 , λ a I Ω IJ ∂ J = 0 , The number of independent null generalized vectors N depends on the choice of the section.
As was shown in [24], N cannot be greater than d, but we can always choose N = d or N = d − 1, which correspond to the M-theory section and the type IIB section, respectively.
In fact, the M-theory section and the type IIB section can be described by the following set of null vectors,λ a (N = d) andλ a (N = d − 1): In the former case,λ a η k ∂ = 0 andλ a η k 1 ···k 4 ∂ = 0 require ∂ i 1 i 2 = ∂ i 1 ···i 5 = 0 . On the other hand,λ a η k 1 ···k 7 , l 1 l 2 l 3 ∂ = 0 andλ a η k 1 ···k 7 , l 1 ···l 6 ∂ = 0 are trivially satisfied. The remaining conditions,λ a η k 1 ···k 6 , l ∂ = 0 andλ a Ω ∂ = 0, require ∂ i 1 ···i 7 , i = 0. Therefore,λ a describes the M-theory section where all fields depend only on x i . The quadratic section condition η IJ; I ∂ I ⊗ ∂ J = 0 is trivially satisfied on this section. Similarly, in the latter case, we can easily show that all fields depend only on x m , andλ a describes the type IIB section.
In order to describe a more general section, we can rotate the above canonical sections,λ a andλ a , by U -duality transformations: Since η IJ; I and η MN; M behave as the Clebsch-Gordan-Wigner coefficients, they satisfy and from this, we can easily show η IJ; which are equivalent to the quadratic section condition. In an example of the E 6(6) EFT, the null conditions for the generalized vectors, (λ a If λ a i is invertible, we can choose λ a i = δ a i and then the first equation requires λ ka; b = −λ kb; a . Then, we can express λ ij; k by using a 3-vector (i.e. antisymmetric third-rank tensor) ω ijk , λ ij; k = −ω ijk , and the second equation becomes (5.14) This leads to where ω k 1 ···k 6 is an arbitrary 6-vector. The last equation in (5.13) is trivially satisfied. Therefore, the most general parameterization is given by (5.16) In this way, when λ a i is invertible, the most general parameterization of λ a is obtained fromλ a via a U -duality transformation generated only by negative-root generators R i 1 i 2 i 3 and R i 1 ···i 6 (GL(d) generators K i j are not necessary). 2 It is also the case for lower exceptional groups d ≤ 5 . The same will be the case for E 7(7) also, and in that case, λ a will be specified by 42 (= 35 + 7) parameters ω i 1 i 2 i 3 and ω i 1 ···i 6 . Similarly, in the case of the type IIB section, if λ a m is invertible, the most general parameterization ofλ a will be given bȳ λ a = e
On the other hand, the linear section equation is expressed as [11] Λ [a ∂ bc] = 0 (a, b, c = 1, . . . , 5) , where Λ a are arbitrary parameters that specify the section (which is considered to be a generalized notion of the pure spinor that specifies a generalized notion of the Dirac structure [11]  where λ a If we consider a case where λ a k is invertible, we can choose λ a k = δ a k and the first equation shows λ ij; a = −ω ija with ω ijk = ω [ijk] . The second equation is then automatically satisfied since the following identity is satisfied in d = 4: Therefore, the set of the null vectors becomes The linear section equations λ a I η IJ; k ∂ J = 0 and λ a I η IJ; k 1 ···k 4 ∂ J = 0 then become The first condition is precisely the second equation in (5.20) if we make the identifications The second condition follows from the first. In this sense, when λ a i is invertible, our linear section equations are equivalent to (5.19).
For completeness, let us see the number of independent parameters that specify a section.
The linear section equation (5.19) includes 5 parameters Λ a , but as we can see from (5.20), only the 4 ratios Λ i /Λ 5 specify the section. This matches with the number of independent parameters ω ijk entering in our section equations.
If we consider a case where λ a i is not invertible, we may find an inequivalent section. For example, when (λ a i ) = diag(1, 1, 0, 0), (λ 34; 3 ) = 1, and other components vanish (λ 4 is a zerovector in this case), the null condition is trivially satisfied, and the linear section equation shows fields can depend only on x 1 , x 2 , and y 34 . This is the well-known type IIB section considered in the next section with a different parameterization of the generalized coordinates.
Note that if the number of non-vanishing components of λ a is too small, the linear section equation is not sufficient to reproduce the section condition.
The generalized coordinates x ab in the literature are related to our generalized coordinates as follows, and the coordinates {x 15 , x 25 , x 12 } correspond to the physical coordinates {x 1 , x 2 , x 3 } in the type IIB parameterization: x Our linear section equations are specified bȳ These are precisely equations (5.30) if we make the following identifications: In this sense, our linear section equations are equivalent to the linear section equation (5.28) for the type IIB section in the SL(5) EFT.

Conclusions and discussion
In this paper, we obtained a set of η-symbols associated with branes in the string multiplet, and 168 1 (2,5,1) are dual to each other [29].
Our considerations are limited to d ≤ 7, but we can also consider the E 8 (8) EFT, where the number of η-symbols will be the same as the dimension of the R 2 -representation of E 8(8) , namely 3875. According to [29], the branes in the string multiplet can be summarized as in Table 1. There, each brane in the table is wrapping a certain cycle in the 8-torus T 8 and behaves as a string with a tension T in the uncompactified spacetime. We call the brane a "b (c,d,e) -brane" if the tension of the string takes the form where R i denotes the radius along the x i -direction (i = 1, . . . , 8), and ℓ 11 is the 11-dimensional Planck length. We also define b (d,e) ≡ b (0,d,e) and b e ≡ b (0,e) . It will be interesting to determine all of the η-symbols associated with the 3875 branes.
In this paper, we have not discussed the role of the η-symbols in worldvolume theories in detail, but in fact, they play an important role. In the T -duality manifest formulation of the string, the equations of motion can be expressed as the self-duality relation [44], where * γ is the Hodge star operator on the worldsheet associated with the metric γ, and As a generalization of this relation, if we consider a membrane theory in the approach of [45], the equations of motion can be expressed as by using the η-symbol η k associated with an M2-brane. It will be interesting to see whether this kind of self-duality relation is satisfied for all of the branes in the string multiplet. It is also interesting to see how the Ω-tensor appears in the brane worldvolume theories.

A Conventions and formulas A.1 Combinatoric factors
We shall use the following convention for multiple indices. When we consider M-theory, the generalized vector is parameterized as The combinatoric factors are introduced such that the indices are summed with weight 1 when we consider the ordered multiple indices i 1 · · · i p , which satisfy i 1 < · · · < i p . For example, the inner product between V I and W I becomes and in the second line, all components are summed with weight 1. Similarly, the generalized coordinates and derivatives are defined as We define the derivative as which gives, e.g., ∂ 12 y 12 = 1/2 and ∂ 12 y 12 = 1 . If we define the Kronecker delta as they satisfy If we use the ordered indices, e.g., the matrix η k 1 ···k 4 has a simpler form. Indeed, the complicated numerical factors disappear: In fact, all of the η-symbols except those associated with KKM and 8-branes (or KKM and 7 2 -branes in the type IIB side) have a simple form without complicated numerical factors.
If we stick to the unordered multiple indices, as in the main text, the rule for the numerical factor is as follows: For a {i 1 · · · i p , k 1 · · · k q }-{j 1 · · · j r , l 1 · · · l s } component of η I , we introduce

A.2 E d(d) group
The simple roots of the E d(d) group are denoted by α n (n = 1, . . . , d) and their relation is shown in the following Dynkin diagram: In this convention, the R 1 -/R 2 -representations are defined by the following Dynkin labels: The generators of the E d(d) group (d ≤ 7) can be parameterized in two different ways, depending on whether we are considering M-theory or type IIB theory [10,18,46,47]: where i, j = 1, . . . , d, m, n = 1, . . . , d − 1, and α, β = 1, 2 . By considering R αβ = R (αβ) , the number of the above generators is the same as the dimension of the E d(d) group (d ≤ 7).
In the M-theory parameterization, the explicit forms of the generators are given by On the other hand, in the type IIB parameterization, the explicit forms of the generators are given by (A.23)

B Comparison with known Y -tensors
In this appendix, we reproduce known Y -tensors from our result.

B.1 Y -tensor in SL(5) EFT
In the SL(5) EFT, we have 5 non-vanishing η-symbols, which can be redefined as If we redefine the coordinates as ( These can be neatly summarized as follows by introducing indices a, b, c = 1, . . . , 5 and a totally antisymmetric tensor ǫ a 1 ···a 5 satisfying ǫ i 1 ···i 4 5 = ǫ i 1 ···i 4 : By further using the conventional parameterization x I ≡ x a 1 a 2 (x i5 ≡ x i ), they become We also define ǫ c = ǫ c IJ = ǫ ca 1 a 2 b 1 b 2 √ 2! 2! , and then Y IJ KL becomes which is summarized as Y IJ KL = ǫ eIJ ǫ eKL in (1.6).
B.2 Y -tensor in SO(5, 5) EFT In the SO(5, 5) EFT, we have 10 η-symbols, which can be redefined as In the coordinates (x I ) = x i , We also define matrices or more explicitly, If we further define which satisfy the relation γ AγB + γ BγA I J = 2 η AB δ I J , (B.13) the Y -tensor can be expressed in the conventional form (1.6), (B.14) B.3 Y -tensor in E 6(6) EFT In the E 6(6) EFT, we have 27 η-symbols, which can be redefined as In the coordinates x i , y i 1 i 2 √ 2 , z i with z i ≡ − 1 5! ǫ ij 1 ···j 5 y j 1 ···j 5 , the matrices become

B.4 Y -tensor in E 7(7) EFT
In the E 7(7) case, we have 133 η-symbols, which can be redefined as  where the above matrices take the following forms: From these matrices, the Y -tensor and the generalized Lie derivative have been explicitly computed in [38].